Fluid substance dispensing pumps are known for example from U.S. Pat. No. 6,170,713. The pump described in said US document comprises a cup-shaped body closed at one end by an endpiece provided with a passage for a stem on which a movable piston is directly provided sealing against the inner surface of the cup-shaped body. The cup-shaped body comprises unidirectional valve means arranged to intercept a fluid passage provided in the base of the cup-shaped body.
A first spring is provided within the cup-shaped body to urge the piston into a rest position. The stem is provided internally with a large axial cavity housing a valving element movable against a sealing surface in opposition to a second spring. This valving element opens to enable the fluid compressed by the piston to be delivered into the cup-shaped body only when the internal fluid pressure exceeds that with which the spring holds the valving element pressed against a sealing surface provided in the stem.